LemmaStatement: AI-generatedProof: AI-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Products and projections on image filtrations

PDF p. 10: Equation (4.3)

Statement

Let $F$ be a multiplicative decreasing filtration on a commutative ring spectrum $E$ . For integers $i, j, a, b$ , write $q_{i, a} : π_{a} F^{i} ⟶ π_{a} E, p_{i, a} : π_{a} F^{i} ⟶ π_{a} {gr}^{i} F$ for the maps induced by the augmentation and the cofiber projection, respectively, and set $F^{i} π_{a} E = im q_{i, a}$ . Then $F^{i} π_{a} E \cdot F^{j} π_{b} E \subseteq F^{i + j} π_{a + b} E .$ For any stage lifts $x \in π_{a} F^{i}$ and $y \in π_{b} F^{j}$ , their stage product $x y \in π_{a + b} F^{i + j}$ satisfies $p_{i + j, a + b} (x y) = p_{i, a} (x) p_{j, b} (y) .$ If, for the three pairs $(i, a)$ , $(j, b)$ and $(i + j, a + b)$ , one has $\ker q_{r, n} \subseteq \ker p_{r, n}$ , the corresponding maps $p_{r, n}$ descend uniquely to homomorphisms ${\overset{ˉ}{p}}_{r, n} : F^{r} π_{n} E \to π_{n} {gr}^{r} F$ , and ${\overset{ˉ}{p}}_{i + j, a + b} (u v) = {\overset{ˉ}{p}}_{i, a} (u) {\overset{ˉ}{p}}_{j, b} (v)$ for $u \in F^{i} π_{a} E$ and $v \in F^{j} π_{b} E$ . Injectivity of each relevant $q_{r, n}$ is sufficient for these descent conditions; it is not needed for the containment or the formula for chosen stage lifts.

A decreasing filtration here is a coherent tower of spectra $F^{r}$ with maps $F^{r + 1} \to F^{r}$ and compatible maps $F^{r} \to E$ . Its graded piece is the cofiber ${gr}^{r} F = cofib (F^{r + 1} \to F^{r})$ . Multiplicativity supplies compatible maps $F^{i} \land F^{j} \to F^{i + j}$ and ${gr}^{i} F \land {gr}^{j} F \to {gr}^{i + j}$ , and makes the augmentation compatible with multiplication on $E$ . Products in homotopy groups are those induced by these maps and the canonical smash pairing of sphere spectra. An image filtration consists of the subgroups $F^{r} π_{n} E$ just defined; a stage lift of an element of one of these subgroups is a preimage under $q_{r, n}$ .

Facts & Assumptions

Given: A multiplicative decreasing filtration $F$ on a commutative ring spectrum $E$ and integers $i, j, a, b$ .

Goal: Prove the product containment and the projection identity for stage lifts, and prove the stated conditional identity on image groups.

[F1]

The stage multiplication is compatible with the augmentation to $E$ , and the graded multiplication satisfies ${\overline{μ}}_{i, j} (p_{i} \land p_{j}) ≃ p_{i + j} μ_{i, j}$ for the cofiber maps $p_{r} : F^{r} \to {gr}^{r} F$ . These maps induce the corresponding pairings on homotopy groups. Multiplicative filtrations and graded products

[F2]

For each $(r, n)$ , the stage projection $p_{r, n}$ factors uniquely through the image of $q_{r, n}$ exactly when $\ker q_{r, n} \subseteq \ker p_{r, n}$ . In particular, injectivity of $q_{r, n}$ implies this condition. Stage projections, kernels and image groups

Proof

technique · Apply the augmentation and cofiber compatibilities to representatives, then use the image-factorization criterion

1.1F1givenconstruct

Let $u \in F^{i} π_{a} E$ and $v \in F^{j} π_{b} E$ . By the definition of image, choose $x \in π_{a} F^{i}$ and $y \in π_{b} F^{j}$ with $q_{i, a} (x) = u$ and $q_{j, b} (y) = v$ . The stage multiplication gives $x y \in π_{a + b} F^{i + j}$ . Compatibility of multiplication with the augmentation gives $q_{i + j, a + b} (x y) = u v$ . Hence $u v$ is in $F^{i + j} π_{a + b} E$ , proving the containment.

1.2F1given

For any such stage classes $x, y$ , the cofiber compatibility in [F1], applied to their representing sphere maps, gives $p_{i + j, a + b} (x y) = p_{i, a} (x) p_{j, b} (y)$ . This equality concerns the chosen stage classes and does not require their images in $π_{*} E$ to determine them uniquely.

2.1F2step 1.1step 1.2assume-hyp

Assume the three kernel inclusions in the statement. By [F2], the three stage projections factor uniquely through their respective image groups. For the lifts chosen in step 1.1, the product $x y$ is a stage lift of $u v$ . Therefore ${\overset{ˉ}{p}}_{i + j, a + b} (u v) = p_{i + j, a + b} (x y) = p_{i, a} (x) p_{j, b} (y) = {\overset{ˉ}{p}}_{i, a} (u) {\overset{ˉ}{p}}_{j, b} (v) .$ The factorization ensures that the displayed values are independent of the chosen lifts.

3.1F2step 1.1step 1.2step 2.1∎

An injective $q_{r, n}$ has zero kernel and thus satisfies the corresponding inclusion. Steps 1.1 through 2.1 prove all assertions, with no injectivity hypothesis used in the first two steps.

Depends on

Used by · 1

Filtration projections and multiplication

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